limit x->1 (x^2-1)/(x-1) — step-by-step solution

1) Let's evaluate the limit of

\frac{x^{2} - 1}{x - 1}

when

x \to 1

ⓘ The power with a constant exponent is continuous here: substitute the limit of the base

The limit of

x^{2}

1

when

x \to 1

ⓘ The limit of a sum of finite limits is the sum of the limits

The limit of

x^{2} - 1

1 - 1

when

x \to 1

⇥1.1) Let's calculate:

1 - 1

⇥ⓘ Start from

1

and move back

1

on the number line, down to zero:

1 - 1

0

⇥

0

ⓘ The limit of a sum of finite limits is the sum of the limits

The limit of

x - 1

1 - 1

when

x \to 1

⇥1.2) Let's calculate:

1 - 1

⇥ⓘ Start from

1

and move back

1

on the number line, down to zero:

1 - 1

0

⇥

0

For

\frac{x^{2} - 1}{x - 1}

, we encounter the indeterminate form 0/0 as

x

tends to

1

2) To find the limit, we factorize the numerator and the denominator:

x^{2} - 1

For

x^{2} - 1

, we recognize a difference of two squares:

x^{2} - 1^{2}

. Apply

a^{2} - b^{2} = (a - b) \cdot (a + b)

, with

a = x

and

b = 1

Factors found:

(x + 1) \cdot (x - 1)

3) Let's reduce:

\frac{(x + 1) \cdot (x - 1)}{x - 1}

ⓘ The original denominator must remain nonzero. The fraction is simplified by dividing the numerator and the denominator by their common factor

x - 1

\frac{(x + 1) \cdot (x - 1)}{x - 1}

x + 1

x + 1

ⓘ The limit of a sum of finite limits is the sum of the limits

The limit of

x + 1

1 + 1

when

x \to 1

Let's evaluate:

1 + 1

ⓘ Start from

1

and move forward

1

on the number line:

1 + 1

2

2

So the limit of

\frac{x^{2} - 1}{x - 1}

2

when

x \to 1

▶Limit:

2